Putting Reasoning and Sense Making at the Center

2. Putting Reasoningand Sense Making at the Center Focusin High School Mathematics

3. Two Classrooms In Ms. Stree’s class, the following question was asked: • The driving distance Boston to Chicago is 990 miles. Rico drives from Boston to Chicago at an average speed of 50 mph and returns at an average speed of 60 mph. For how many hours is Rico on the road? Graham, K., Cuoco, A., & Zimmermann, G. (2010). Focus in High School Mathematics: Reasoning and Sense Making in Algebra. Reston, VA: NCTM.

4. Two Classrooms In Ms. Taque’s class, this question was asked: • Rico drives from Boston to Chicago at an average rate of 50 mph and returns by the same route at an average speed of 60 mph. If he is on the road for 36 hours, how far is it from Boston to Chicago? Graham, K., Cuoco, A., & Zimmermann, G. (2010). Focus in High School Mathematics: Reasoning and Sense Making in Algebra. Reston, VA: NCTM.

5. Two Classrooms The two questions show the divide between what happens in typical classrooms. Are instructors asking students to reason with mathematics and reason through problems? Or are teachers giving prompts that require little engagement and reasoning? Graham, K., Cuoco, A., & Zimmermann, G. (2010). Focus in High School Mathematics: Reasoning and Sense Making in Algebra. Reston, VA: NCTM.

6. History of NCTM Standards 2000 - Principles and Standards for School Mathematics • Updated the 1989 standards, incorporating • Professional Standards for Teaching Mathematics (1991) • Curriculum and Evaluation Standards for School Mathematics (1995)

7. History of NCTM Standards 2006 - Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence • The most important mathematical topics for each grade level, based on Principles and Standards But what about high school mathematics?

8. Reasoning and Sense Making A focus on reasoning and sense making, when developed in the context of important content, will ensure that students can accurately carry out mathematical procedures, understand why those procedures work, and know how they might be used and their results interpreted. National Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA, p.3

9. Reasoning and Sense Making • Reasoning is not a set of topics but a stance toward learning mathematics • Reasoning and sense making throughout the curriculum lends coherence across domains: number, algebra, geometry, and statistics • Emphasizes underlying connections • Connections, in turn, promote reasoning and sense making that strengthens coherence, allowing streamlining of the curriculum

10. Reasoning and Sense Making Reasoning and sense making is a way to approach instruction no matter what content you are teaching.

11. What Is Reasoning and Sense Making? • Reasoning: The process of drawing conclusions on the basis of evidence or stated assumptions. • Sense making: Developing understanding of a situation, context, or concept by connecting it with existing knowledge.

12. Reasoning and Sense Making Reasoning and sense making is an evolution of NCTM’s longstanding position that problem solving should be the emphasis of mathematics teaching and learning.

13. Reasoning and Sense Making “The processes of mathematics— Problem Solving, Reasoning and Proof, Connections, Communication, and Representation—are all manifestations of the act of making sense and of reasoning …” National Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA, p.5

14. Reasoning and Sense Making • Levels of reasoning • Informal explanation • Inductive observations • Justification • Formal deduction

15. Reasoning and Sense Making Reasoning is the foundation of mathematical competence • Conceptual understanding • Procedural fluency • Strategic competence • Adaptive reasoning • Productive disposition National Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA, p.12

16. Reasoning andSense Makingshould occur every day in the high school mathematics classroom

17. The Examples in Focus in High School Mathematics • Demonstrate how reasoning and sense making can be incorporated into the high school curriculum. • Paint an idealized picture of what reasoning and sense making should look like in practice. • Take a variety of formats.

18. Reasoning and Sense Making Focus in High School Mathematics highlights reasoning opportunities in five content areas: • Number and Measurement • Algebraic Symbols • Functions • Geometry • Statistics and Probability

19. Using Tasks Fuel for Thought A teacher gives her students the following prompt taken from an article in the New York Times (Chang 2008) and asks them to explain their reasoning.

20. Fuel for Thought Which of the following would save more fuel? Replacing a compact car that gets 34 miles per gallon (MPG) with a hybrid that gets 54 MPG Replacing a sport utility vehicle (SUV) that gets 18 MPG with a sedan that gets 28 MPG Both changes save the same amount of fuel.

21. Fuel for Thought Discuss with a small group possible solutionsfor this problem.

22. Fuel for Thought Share your solutions

23. Fuel for Thought Sample response: I see that the change from 34 to 54 MPG is an increase of 20 MPG, but the 18 to 28 MPG change is only a change of 10 MPG. So, replacing the compact car saves more fuel.

24. Fuel for Thought Sample response: The change from 34 MPG to 54 MPG is an increase of about 59% while the change from 18 to 28 MPG is an increase of only 56%. So the compact car is a better choice.

25. Fuel for Thought Sample response: I thought about how much gas it would take to make a 100-mile trip. Compact car: 100 miles/54MPG = 1.85 gallons used 100 miles/34MPG = 2.94 gallons used SUV: 100 miles/28MPG = 3.57 gallons used 100 miles/18MPG = 5.56 gallons used

26. Fuel for Thought The compact car saved 1.09 gallons while the SUV saved 1.99 gallons for every 100 miles. That means you actually save more gasoline by replacing the SUV.

27. Reasoning and Sense Making Technology • Technology can be used to advance the goals of reasoning and sense making • Technological tools can relieve students of burdensome calculations and can facilitate the search for patterns and relationships and the formation of conjectures.

28. Fuel for Thought

29. Using Tasks Fuel for Thought • Key Mathematical Elements • Number and Measurement (reasonableness of answers and measurements) • Functions (multiple representations of functions) • Reasoning Habits • Analyzing a problem (seeking patterns and relationships) • Reflecting on a solution (interpreting a solution; reconciling different approaches; refining arguments)

30. Strategy: Multiple Entry Points Use problems that have multiple entry points so that students at different levels of mathematical experience and with different interests can all engage meaningfully in reasoning about a problem.

31. How can you get started? • Think about any standard topic in your curriculum. • Recast the content as questions that students can explore • Resist the temptation to tell students the content - Believe that students can investigate and derive relationships and mathematical concepts.

32. How can you get started? Consider finding x- and y-intercepts of a linear function What ways do you currently use to teach this?

33. How can you get started? Finding x- and y-intercepts • Ask students to look at a graph and identify the intercepts • Ask students why these points may be of interest to someone trying to interpret the graph • Create a context for a graph that requires interpretation of one of the intercepts

34. How can you get started? Finding x- and y-intercepts • Ask students to create their own rules and to compare these to the textbook rules. Why do the rules work? • Ask students to graph a “messy” equation that’s difficult to graph: 33x + 14y = 231

35. Review of Tips for the Classroom National Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA Provide tasks that require students to figure things out for themselves Ask students to restate problems in their own words Give students time to analyze a problem intuitively, explore the problem further using models, and then proceed to a more formal approach Resist the urge to tell students how to solve problems Ask questions that will press students thinking

36. Review of Tips for the Classroom National Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA Provide adequate wait time Encourage students to ask probing questions of themselves and each other Expect students to communicate their reasoning to their classmates and teacher verbally and in writing Highlight exemplary examples and have students reflect on what makes them effective Establish a classroom climate in which students feel comfortable sharing their mathematical arguments and critiquing the arguments of others

37. Review of Tips for the Classroom • Considering and evaluating alternative explanations • Understanding the allowable scope of conclusions • Determining whether a conclusion based on the context is plausible Shaughnessy, J. M., Chance, B., & Kranendonk, H. (2009). Focus in High School Mathematics: Reasoning and Sense Making in Statistics and Probability. Reston, VA: NCTM.

38. Reasoning and Sense Making Equity • High expectations for all students • Courses offer rich opportunities for reasoning and sense making • Allow all students to see mathematics as important for their lives and future careers

39. Published in 2009

40. This will be followed by a book that offer examples of ways to incorporate technology to promote reasoning and sense making.

41. Focus in High School Mathematics: Reasoning and Sense Making • For more information:www.nctm.org/hsfocus • Materials include an executive summary and brochures for • Teachers • Students • Administrators • Policymakers • Families